Naturally, system (REF ) must be consistent with the kinetic equation. Indeed, kinetic equation (REF ) implies that for any \(h(\eta ) \ne 0\) , \(\int _\Gamma h(\eta )f (\eta ; x, t) \mathrm {d}\eta \)  is a conserved quantity (density) with \(\int _\Gamma h(\eta )f (\eta ; x, t) s(\eta ; x, t)\mathrm {d}\eta \)  being the corresponding flux density. The specific “Kruskal” series in (REF ) is obtained as [1]}
\(\langle P_n[u ] \rangle = C_n\int _\Gamma \eta ^{2n+1} f(\eta ) \mathrm {d}\eta \; , \quad \langle Q_n[u ] \rangle = C_n\int _\Gamma \eta ^{2n+1} f(\eta ) s(\eta ) \mathrm {d}\eta \; ,\) 
